It is relevant, in order to simplify the interpretations, to focus on the \(r_\perp =0\)  axis, since clusters are known to belong to this axis, in axial symmetry (see e.g. [1]}). In this case, only \(m_l=0\)  states remain, and one gets from Eqs. (REF ) and (REF ):
\(\begin{split}Z(z)=\frac{2}{b^4_\perp b^2_z\pi ^2}\sum _{n_z\ne n^{\prime }_z}&\phi _{n^{\prime }_z}\phi _{n_z-1}[n_z\phi _{n^{\prime }_z}\phi _{n_z-1}\\&\quad -\sqrt{n_zn^{\prime }_z}\phi _{n_z}\phi _{n^{\prime }_z-1}]\end{split}\) 
